A rope by which a calf is tied is decreased from 23 m to 12 m. What is the decrease in area to be grazed by it ?

Options

A. `1110 m^(2) `
B. `1210 m^(2) `
C. `1120 m^(2) `
D. `1221 m^(2) `

Answer

To find the decrease in the area that the calf can graze when the length of the rope is decreased from 23 m to 12 m, we will follow these steps:

### Step 1: Calculate the area grazed with the original rope length (23 m)
The area grazed by the calf when tied with a rope of length 23 m forms a circle. The formula for the area of a circle is given by:

\[ \text{Area} = \pi r^2 \]

Here, \( r = 23 \) m.

\[ \text{Area}_1 = \pi (23)^2 = \frac{22}{7} \times 23 \times 23 \]

Calculating \( 23^2 \):

\[ 23^2 = 529 \]

Now substituting back into the area formula:

\[ \text{Area}_1 = \frac{22}{7} \times 529 \]

### Step 2: Calculate the area grazed with the new rope length (12 m)
Now, we will calculate the area grazed when the rope length is decreased to 12 m.

\[ \text{Area}_2 = \pi (12)^2 = \frac{22}{7} \times 12 \times 12 \]

Calculating \( 12^2 \):

\[ 12^2 = 144 \]

Now substituting back into the area formula:

\[ \text{Area}_2 = \frac{22}{7} \times 144 \]

### Step 3: Calculate the decrease in area
To find the decrease in area, we subtract the area grazed with the new rope length from the area grazed with the original rope length:

\[ \text{Decrease in Area} = \text{Area}_1 - \text{Area}_2 \]

Substituting the areas we calculated:

\[ \text{Decrease in Area} = \left(\frac{22}{7} \times 529\right) - \left(\frac{22}{7} \times 144\right) \]

Factoring out \(\frac{22}{7}\):

\[ \text{Decrease in Area} = \frac{22}{7} \left(529 - 144\right) \]

Calculating \( 529 - 144 \):

\[ 529 - 144 = 385 \]

Now substituting back:

\[ \text{Decrease in Area} = \frac{22}{7} \times 385 \]

### Step 4: Calculate the final decrease in area
Now we will calculate:

\[ \text{Decrease in Area} = \frac{22 \times 385}{7} \]

Calculating \( 22 \times 385 \):

\[ 22 \times 385 = 8470 \]

Now dividing by 7:

\[ \text{Decrease in Area} = \frac{8470}{7} = 1210 \text{ m}^2 \]

### Final Answer
The decrease in the area that the calf can graze is **1210 m²**.

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